Make a mesh/plane deform on collision (and stay deformed)
As suggested by timaroberts shapekeys could be a way to achieve this.

Shape keys a are a way to store or save a geometry state of you mesh, allowing to animate by morphing or blending between them freely.


Add a new plane object
Subdivide it as much as you need
Exit edit mode and go to the Properties Window > Object Data > Shape Keys
Add two shapekeys (base and one to make your deformation)
Select the second key
Enter edit mode
Turn on proportional edit with O key (letter "o" not zero) 
Grab the central faces or vertex and move them down in Z axis
Use Mouse Wheel Scroll to adjust proportional editing size accordingly
Exit edit mode and you can blend freely between shapes.